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2. Probabilities with Multiple Events
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Pre-Algebra /
Chapter 12
2. 

Probabilities with Multiple Events

Understanding probabilities with multiple events is essential for analyzing situations involving uncertainty. This lesson introduces sample spaces and tree diagrams to visualize possible outcomes. It also explores compound events, which involve multiple conditions or scenarios, and demonstrates how simulations can be used to model and analyze probabilities in real-world contexts. By mastering these concepts, students can solve complex probability problems and apply these skills to fields like statistics, science, and decision-making.
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Probabilities with Multiple Events
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Events are typically not isolated, and they can occur along with other events. The concept of compound event is used to understand the odds of two or more events happening together. Compound events are used to describe a wider variety of scenarios using the basics of probability. This lesson will discuss compound events and some tools that are useful when studying them.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Getting the Right Coffee

One particular Sunday morning, Magdalena went out with her parents to do the groceries. On their way back to home, they stopped at a drive-thru coffee shop to get some drinks. As they were on the ordering screen, Magdalena got a text from her sister, asking her for a coffee. Unfortunately, she did not elaborate on how she would like her coffee.

Magdalena's sister asking for a coffee in a text message
Magdalena knows her sister always asks for an American. However, she does not know if she likes it black, or if she puts any kind of milk in it. On top of that, she does not know whether to sweeten it or not! Since they are already in line, and there are people waiting behind them, Magdalena is short on time. Magdalena will have to make a guess on how her sister would like her coffee.
Ordering screen applet asking for the details about the american coffee Magdalena is about to order
There are two options for the coffee, normal and decaf. It is also possible to add either whole, skim, soy, or almond milk (or no milk at all). Finally, it can be sweetened with regular sugar, raw sugar, stevia, or unsweetened. What are the chances that Magdalena randomly orders the coffee exactly as her sister would have liked?
Discussion

Sample Space

The sample space of an experiment is the set of all possible outcomes. For example, when flipping a coin, there are two possible outcomes: heads, H, or tails, T. Therefore, the sample space is

Coin flip outcomes: Head and Tail in the sample space.
Example

Picking the Ingredients for a Sandwich

Magdalena and her family have a very diverse taste on sandwiches. For example, Magdalena likes a simple ham sandwich made with white bread. On the other hand, her dad likes a roast beef rye sandwich. Because of this, they usually buy three different types of deli meat: roast beef, ham, and turkey. Deli compressed.JPG On the bread side, everyone likes white bread aside from her dad, who prefers rye bread. Magdalena wonders how many different sandwiches can be made from these ingredients. Bread compressed.jpg Write the sample space of all the possible sandwiches that can be made using white bread, rye bread, roast beef, ham, or turkey. Assume that only one type of deli meat can be used on each. How many different sandwiches can be made?

Hint

Use different combinations of letters to represent each type of sandwich. For example, let WR be a roast beef sandwich made with white bread. Count how many different combinations can be written.

Solution
There are two different bread options for the sandwich, white bread an rye bread. The first letter of each type of bread can be used to represent which one will be used in a sandwich.
Similarly, the first letter of each type of deli meat can be used.
Notice that the letter R is used both for rye bread and for roast beef. This is not a problem at all as long as it is understood that the first letter corresponds to the bread, and the second letter to the deli meat. Knowing this, the sample space can now be written. Keep in mind that no deli meat is to be repeated.
Now that the sample space has been written, the number of elements can be counted. There are a total of different sandwiches that can be made from the given options of bread and deli meat!
Discussion

Tree Diagrams

A tree diagram is an illustration of all the possible combinations that can be made from a given set of objects. Tree diagrams are useful for obtaining the sample space of an experiment when multiple choices are available. As an example, consider the different pasta recipes that can be made with different types of pasta, sauce, and cheese.

Tree diagram
In the above diagram, there are different types of pastas, different sauces, and different cheeses. A tree diagram illustrates the sample space containing all different options that can be made by combining the given ingredients.
Example

Adding Cheese to the Sandwiches

Magdalena's family also likes different types of cheese. From the wide variety of existing cheese available, they typically buy American, Swizz, and provolone. Cheese shop compressed.jpg Magdalena now wonders how many different sandwiches can be made using the different types of bread, deli meat, and cheese. Assume that only one type of deli meat and cheese can be used. Use a tree diagram to find how many different sandwiches can be made from the above ingredients.

Hint

There are three categories to consider when making the sandwich: bread, deli meat, and cheese. When moving to the next category, all the options need to be repeated.

Solution

Adding cheese as an ingredient for the sandwiches makes the task of finding the sample space a little harder. A tree diagram can provide a useful visual aid. Use the tree diagram to find how many different combinations of bread, meat, and cheese can be made from the given options. Begin by listing the two bread options.

Making the tree diagram. The first row from left to right: Bread, Meat, Cheese, Sample Space. The bread column lists White and Rye.

Next, list the three options for the deli meat next to each bread option. Use segments to connect both bread types to each of the deli meats available.

Making the tree diagram. The first row from left to right: Bread, Meat, Cheese, Sample Space. The bread column lists White and Rye. The meat columns lists Roast Beef, Ham, Turkey twice, for each bread.

Similarly, next to each meat option list the three cheese options. Draw segments connecting meat options to cheese options.

Making the tree diagram. The first row from left to right: Bread, Meat, Cheese, Sample Space. The bread column lists White and Rye. The meat column lists Roast Beef, Ham, Turkey twice, for each kind of bread. The cheese column lists American, Swizz, Provolone three times, for each kind of meat.

Each possible outcome is represented by a path in the tree diagram. Use the first letter of each ingredient to represent all the possible sandwiches that can be made with the given ingredients. This way, the sample space can be obtained.

Tree diagram
The sample space consists of elements. This means that different sandwiches can be made with the given ingredients. Notice that the same result can be found by multiplying the number of options in each category. There are different types of bread, different deli meats, and different cheeses.
Discussion

Compound Event

A compound event is an event that is a combination of two or more events. Below is an example.
The number of elements of the sample space of the compound event can be found by multiplying the number of possible outcomes of each individual event.
Event Number of Outcomes
Flipping a coin
Rolling a die
Flipping a coin and rolling a die

In the above example, possible outcome of the compound event is flipping a tails and rolling a To find the probability of this outcome, divide this one possibility by the number of possible outcomes of the compound event.

Example

Will Mom Like the Sandwich?

Magdalena knows that her mom loves provolone cheese. She even usually eats some slices of ham with olives as a snack! Magdalena now asks her mom what is her favorite type of meat.

Magdalena having a conversation with her mom about how she likes her sandwiches. Magdalena's mom says she just does not like ham.

Magdalena recalled that only her dad likes rye bread, so her mom should like white bread. Magdalena decided to use the tree diagram she made when exploring all the possible sandwiches.

Tree diagram listing all the possible options that can be made using wite and rye bread; roast beef, ham, and turkey; and american, swizz, and provolone cheese
What is the probability that Magdalena's mom likes a sandwich made with random ingredients?

Hint

Divide the number of favorable outcomes by the total number of outcomes.

Solution

Consider the given tree diagram.

Tree diagram listing all the possible options that can be made using wite and rye bread; roast beef, ham, and turkey; and american, swizz, and provolone cheese

To find the probability that Magdalena's mom likes a sandwich made from random ingredients, the number of favorable outcomes needs to be divided by the total number of outcomes. Magdalena's mom prefers white bread.

Tree diagram listing all the possible options that can be made using wite and rye bread; roast beef, ham, and turkey; and american, swizz, and provolone cheese. The white bread option is highlighted.

From here, the meat can be either roast beef or turkey. Highlight these options in the tree diagram.

Tree diagram listing all the possible options that can be made using wite and rye bread; roast beef, ham, and turkey; and american, swizz, and provolone cheese. Highlight on white bread followed by roast beef and turkey options.

From both of these points, continue to the branches that contain provolone cheese.

Tree diagram listing all the possible options that can be made using wite and rye bread; roast beef, ham, and turkey; and american, swizz, and provolone cheese. Highlight on white bread followed by roast beef provolone, and turkey provolone.
There are favorable outcomes out of the possible outcomes. Find the probability by doing the division.
Notice that the above fraction can be simplified.
Rewrite

Rewrite as

CancelCommonFac

Cancel out common factors

SimpQuot

Simplify quotient

This means that the probability that Magdalena's mom will like a sandwich made at random from the given ingredients is
Discussion

Simulation

A simulation is a model that imitates a real-life process or situation. In particular, a simulation can be used as a probability model to make predictions about real-life events. Consequently, the experimental probability can be estimated by simulating the events.
In general, simulations are used when actual trials of some experiment are impossible or unreasonable to conduct.
Example

The Crane Game

As Magdalena and her parents were leaving from the groceries store, Magdalena saw a crane game filled with cute plush dolls. Crane game compressed.jpg Playing the crane game costs and Magdalena has so she asks her dad if she can play. Magdalena's dad says that the odds of winning a plush doll are about in so he says that it is not worth it.

As soon as Magdalena is in her home, she goes to her laptop and finds an online simulator. In this case, each crane game attempt is simulated as a number between and Since the odds of winning are one out of ten, then one out of ten numbers will represent a success.
Since Magdalena had five random numbers will be generated in succession. To obtain more relevant results, the experiment will be repeated times. Imagine spending that much money on a crane game! Magdalena now runs the simulation and obtains the following numbers.
Using the data from the simulation, what is the experimental probability of getting at least one plush doll when spending to play the crane game?

Hint

Identify which outcomes of the experiment contain at least one success. Divide the number of successes by the total number of trials.

Solution
Since a represents getting a plush doll from the crane game, begin by identifying which trials contain at least one
There were successes out of the trials that were simulated. Find the experimental probability of getting at least one plush doll from the crane game by dividing the number of successes by the number of trials.
Notice that the above fraction can be simplified.
Rewrite

Rewrite as

CancelCommonFac

Cancel out common factors

SimpQuot

Simplify quotient

The simulation showed that the experimental probability of getting at least one plush doll from the crane game is This probability can be written as a percentage.
Keep in mind that by doing a different simulation different probabilities can be found.
Closure

A Lucky Coffee

Magdalena was previously found in the predicament of having people waiting on the line of the drive-thru coffee while not knowing exactly what her sister wanted.
Ordering screen applet asking for the details about the american coffee Magdalena is about to order
A tree diagram can be used to find how many different orders are possible. Since there are plenty of options, the diagram will be huge. Instead, consider that there are options for the type of coffee, options for the type of milk, and options for the sweetener. The number of elements in the sample space can be found by multiplying the number of available options.
This means that Magdalena only has a in chance of getting her Sister's coffee the exact way she likes it.
Magdalena decided to order it regular, no milk, with stevia. It was a big surprise finding out that this is exactly how Magadelena's sister likes it! It is more likely that Magdalena unconsciously recalled how her sister orders her coffee. Anyways, we can say this was a very lucky coffee!
Level 1
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Probabilities with Multiple Events
Exercise 1.1
There are plenty of board games that use eight-sided dice to be played. The faces of this type of die range from to D8 compressed.jpg Write the sample space of the possible outcomes of throwing an eight-sided die.

We are told that an eight-sided die has eight faces. A number from 1 to 8 is written on each face, so there are eight possible outcomes. Write the sample space by listing every possibility. {1, 2, 3, 4, 5, 6, 7, 8} Keep in mind that, if the dice is fair, every outcome is equally as likely as the others.

A local Italian restaurant likes to keep their menu simple. They offer three different pizza flavors: pepperoni, salami, and anchovies. They also have three different sizes: individual, medium, and familiar. Write the number of elements in the sample space corresponding to the different pizzas that can be ordered in the restaurant.

We are asked to write the number of elements in the sample space corresponding to the different pizzas that can be ordered in the restaurant. A tree diagram can help us count these options. Let's begin by writing the different flavors of pizza that the restaurant has to offer.

Next, we list the three options for the size of the pizza next to each flavor. We use segments to connect the flavor and the size of the pizza.

We now write the sample space. We use the first letter of the flavor and the first letter of the size to represent all the possible outcomes.

There are a total of 9 different pizzas that can be ordered in the restaurant.

Suppose a coin is flipped twice. What is the probability of getting heads twice in a row?

We are asked to find the probability of getting heads twice in a row when flipping a coin twice. To do so, we will begin by writing the sample space of the experiment. We can use a tree diagram to help us find the sample space. Let's begin by writing the different outcomes of the first throw.

Next, we list the outcomes of the second throw next to each outcome of the first throw. We use segments to connect both throws.

This lets us write the sample space of the experiment.

Since we are interested in the probability of getting a tails twice in a row we will first identify this outcome in the diagram.

There is only 1 way of getting a tails twice, out of the 4 possible outcomes of the experiment. Finally, we find the probability by dividing the number of favorable outcomes by the total number of outcomes. 1/4 The probability of getting tails twice when flipping a coin is 14 or 25 %.

Suppose a coin is flipped three times in a row. What is the probability of getting exactly two heads?

We are asked to find the probability of getting exactly two heads when flipping a coin three times in a row. To do so we will begin by writing the sample space of the experiment. We can use a tree diagram to help us find the sample space. Let's begin by writing the different outcomes of the first throw.

Next, we write the possible outcomes of the second throw next to the outcomes of the previous throw. We will use segments to connect these throws.

We do the same for the third throw.

Finally, we write the sample space of the experiment.

The sample space consists of 8 elements. Note that we have 3 different ways of getting exactly two heads.

Lastly, we can find the probability of getting exactly two heads by dividing the number of favorable outcomes equal 3 by the number of total outcomes equal 8. 3/8

Probabilities with Multiple Events

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